5.1. The Advance Purchase Discount (APD) Contract
Under the APD contract, firstly, the retailer optimizes preorder quantity
(i.e., prepayment
). Then, the manufacturer optimizes production volume
and carbon emission reduction rate
. The cash flow of the supply chain with the APD contract is as follows (see
Figure 2):
: Retailer pays in advance.
: represents the retailer’s expected sales and his total sales revenue would be . The retailer buys products once more if his preorder could not satisfy market demand. Meanwhile, the retailer’s expected transfer payment is .
: The retailer’s expected inventory is while the manufacturer’s expected inventory is . The per-unit salvage value is . Then, the manufacturer pays carbon emission tax .
Therefore, total profit of retailer will be
In Equation (3), the first term is total expected sales revenue, the second term is total salvage value, the third term is cost of a second order and the last term is a prepayment.
Total profit of manufacturer will be
Equation (4) includes the sales revenue from preselling the sales revenue from second-selling, the total salvage value, the total cost of production and carbon emission tax, as well as the cost of greening technology investment. The manufacturer is required to ensure the preorder quantity. Therefore, her production volume satisfies the constraint .
The advance purchase discount model is a financing method within the supply chain. The manufacturer obtains a short-term loan with an interest rate from the retailer, and she is required to commit a minimum production volume . When the retailer has observed the actual demand , he purchases products with a quantity of at wholesale price and pays the full amount. The manufacturer repays the retailer’s principal and interest . If the manufacturer is unable to repay it, the retailer receives the full revenue of the manufacturer.
According to the dynamic game theory, firstly, the manufacturer’s optimal decisions are solved considering the retailer’s strategy set, and then the retailer’s optimal decisions are solved based on the optimal strategy of the manufacturer [
49]. Therefore, two cases are considered for the retailer’s strategy: (i) the preorder quantity of the retailer is lower than the production volume of the manufacturer, (ii) vice versa. Using the Lagrangian relaxation approach, the manufacturer’s optimal production volume is
and her optimal carbon emission reduction rate is
when
. However, the manufacturer’s optimal production volume is
and her optimal carbon emission reduction rate is
when
. That means the manufacturer’s optimal production volume with the APD contract is
. This result is similar to that of [
19], although his model did not consider carbon emission tax and greening technology investment.
Case 1. The preorder quantity of the retailer is assumed to be less than the production volume of the manufacturer. Substitute into Equation (3) when . Then, we will obtain the optimal preorder quantity of the retailer .
Proposition 3. There exists an advance purchase discount contract . The optimal preorder quantity of the retailer is (i.e., the optimal prepayment of the retailer is ). The manufacturer’s optimal production volume and optimal carbon emission reduction rate are and . Furthermore, the discount factor satisfies the following conditions:, .
Corollary 1. The manufacturer’s optimal production volume and optimal carbon emission reduction rate with the APD contract are equal to those under the decentralized system, i.e., , .
, i.e.,
. It represents that the manufacturer has non-negative profit even if she offers discounts. Thus, the contract
is just a guarantee that the manufacturer will not lose. However, the retailer’s prepayment is not enough to change the manufacturer’s production volume and investment of greening technology
,
, under the contract
. The discount factor
only affects the optimal preorder quantity
and has no influence on the optimal production volume
. If actual demand is less than the production volume of the manufacturer (scenario 1 and scenario 2 in
Table 2), she still needs to bear the inventory risk. If actual demand is less than the preorder quantity of the retailer (scenario 1 in
Table 2), he takes inventory risk. However, if actual demand is higher than the production volume of the manufacturer (scenario 3 in
Table 2), the retailer will not be able to meet demand.
Case 2. The preorder quantity of the retailer is assumed to be greater than the production volume of the manufacturer. Substitute into Equation (3) when . Then, we will obtain the retailer’s optimal decision .
Proposition 4. There exists an advance purchase discount contract . The optimal preorder quantity of the retailer is (i.e., the optimal prepayment of the retailer is ). The manufacturer’s optimal production volume is and her optimal carbon emission reduction rate is . Furthermore, the discount factor satisfies the following conditions: , .
There exists a critical value of the discount factor
.
reflects the manufacturer’s strategy of the APD contract. As the retailer’s retail price decreases, the manufacturer will have a high floor of the discount factor, i.e.,
. The range of the discount factor is shown in
Figure 3 as the carbon emission reduction rate
increases. Further,
Figure 3 shows that the manufacturer’s discount factor
increases with
.
Corollary 2. The optimal preorder quantity of the retailer increases with the discount factor while it decreases with wholesale price, i.e., , .
Corollary 2 suggests that the discount factor affects the optimal preorder quantity . The retailer will increase his preorder quantity when the discount factor increases, in practice, because a high discount factor results in a low order cost for the retailer. However, the higher the wholesale price, the higher the second-order cost of the retailer. Therefore, the retailer prefers to order more products in advance.
Corollary 3. The manufacturer’s optimal production volume and optimal carbon emission reduction rate with the APD contract are greater than those under the decentralized system, i.e., , .
When the discount factor is higher than a certain threshold , the retailer’s prepayment will make the manufacturer’s capacity and investment of greening technology higher than without the prepayment, i.e., , . At this moment, the manufacturer provides a sufficient discount to reduce the retailer’s cost of ordering, which attracts the retailer to purchase more in advance, i.e., . More payment is transferred from the cash on delivery at to prepay at . Meanwhile, the manufacturer has more capital to invest in production and greening technology before the selling season. That means the manufacturer’s cash flow is improved by the prepayment.
For avoiding inventory risks, the manufacturer determines production volume according to the preorder quantity when
, that is,
. If actual demand is lower than the preorder quantity of the retailer (scenario 1 in
Table 3), he bears inventory risk. If actual demand is higher than the preorder quantity of the retailer (scenario 2 in
Table 3), stock shortage risk is also borne by her. Compared with the decentralized system, inventory risk is transferred from manufacturer to retailer. Therefore, from the perspective of avoiding inventory risk, the APD contract benefits the manufacturer.
Proposition 5. , if and only if.
From Proposition 5, we know that if , both the preorder quantity of the retailer and the production volume of the manufacturer are equal to the optimal production volume under the centralized system. Meanwhile, the manufacturer’s optimal carbon emission reduction rate with the contract equals that under the centralized system. Then, supply chain coordination can be reached with the APD contract . The discount factor and the wholesale price are positively correlated under coordination, i.e.,. If the wholesale price increases, the discount factor also increases. In other words, the manufacturer’s net profit on preorders does not increase. Then, the manufacturer only can increase her income by increasing the preorder quantity (i.e., ).
Proposition 6. The optimal carbon emission reduction rate of the manufacturer is increasing in the carbon emission tax, i.e.,. The total profit of the manufacturer is convex in the carbon emission tax .
Substituting
into Equation (4), we have
. When
, the profit of the manufacturer is a decreasing function. When
, the profit of the manufacturer is an increasing function. Therefore, when the carbon emission tax increases, total profit of the manufacturer first decreases and then increases as shown in
Figure 4 and
Figure 5.
5.2. The Option Contract
Under the option contract, firstly, the retailer optimizes option order quantity
. Then, the manufacturer optimizes production volume
and carbon emission reduction rate
. The cash flow of the supply chain with the option contract is as follows (see
Figure 6):
: Retailer pays the option fee in advance.
: represents the retailer’s expected sales and his total sales revenue would be . is the retailer’s cost of exercising options and his expected transfer payment for the second order is .
: is the manufacturer’s total expected salvage value. Then, the manufacturer pays carbon emission tax .
Then, the total profit of the retailer will be
In Equation (6), the first term is sales revenue, the second and third terms are exercising and initial ordering cost of options and the last term is the cost of the second order.
The total profit of the manufacturer will be
In Equation (7), the first, second and third terms are revenues from retailer purchases and exercises of options and second-order purchases, respectively. The fourth term is total salvage value. The fifth term is total cost of production and carbon emission tax. The last term is cost of greening technology investment. The manufacturer needs to ensure the option order quantity. Then, her production volume satisfies the constraint .
Two cases are considered: (i) the option order quantity of the retailer is lower than the production volume of the manufacturer, (ii) vice versa. Using the Lagrangian relaxation approach, the manufacturer’s optimal production volume is
and her optimal carbon emission reduction rate is
when
. However, the manufacturer’s optimal production volume is
and her optimal carbon emission reduction rate is
when
. That means the manufacturer’s optimal production volume with the option contract is
. This result is similar to that of [
49], although their model does not consider the carbon emission tax and greening technology investment.
Case 1. The option order quantity of the retailer is assumed to be less than the production volume of the manufacturer. We obtain the retailer’s optimal option order quantity decision by substituting into Equation (6) when .
Proposition 7. There exists an option contract . The retailer’s optimal option order quantity is . The manufacturer’s optimal production volume is and her optimal carbon emission reduction rate is . Furthermore, the option price satisfies the following conditions:.
Corollary 4. The manufacturer’s optimal production volume and optimal carbon emission reduction rate with the option contract are equal to those under the decentralized system, i.e., , .
If the option price can rise without a cap, the option contract
cannot change the manufacturer’s production volume and investment of greening technology, i.e.,
, . Similar to Proposition 3, the strike price
and the option price
only affect the optimal option order quantity
and have no influence on the optimal production volume
. Regardless of whether actual demand is less or more than the option order quantity of the retailer (scenario 1 and scenario 2 in
Table 4), the manufacturer always needs to bear the inventory risk. When actual demand is higher than the manufacturer’s output (scenario 3 in
Table 4), the retailer will not be able to meet demand.
Case 2. The option order quantity of the retailer is assumed to be greater than the production volume of the manufacturer. We will obtain the retailer’s optimal decision by substituting into Equation (6) when .
Proposition 8. There exists an option contract . The retailer’s optimal option order quantity is . The manufacturer’s optimal production volume is and her optimal carbon emission reduction rate is . Furthermore, the option price satisfies the following conditions: , , .
represents that the manufacturer has non-negative profit with the option contract.
is a critical value of option price.
reflects the manufacturer’s strategy of the option contract. If retail price is high, the manufacturer will have a high cap of the option price, i.e.,
. If strike price is high, the manufacturer will have a low cap of the option price, i.e.,
. Further,
Figure 7 shows that the option price
is decreasing in the carbon emission reduction rate
.
Corollary 5. The optimal option order quantity of the retailer is decreasing in option price and strike price, i.e., , .
Corollary 5 demonstrates that the retailer prefers an option contract with a low option price and a low strike price because it means low risk and low ordering cost. Different from the APD contract
, the optimal option order quantity of the retailer
is independent of the wholesale price
. Since
, the manufacturer can guarantee that the price is favorable to attract the retailer mainly by the option mechanism instead of the instantaneous purchase mechanism [
49]. Hence, the wholesale price does not affect the optimal option order quantity of the retailer.
Corollary 6. The manufacturer’s optimal production volume and optimal carbon emission reduction rate with the option contract are greater than those under the decentralized system, i.e., , .
When the option price is lower than a certain threshold , the retailer’s option fee will make the manufacturer’s production capacity and investment of greening technology higher than without a contract, i.e., , . That means the manufacturer’s cash flow is improved by the option fee. The low option price attracts the retailer to order more options in advance, i.e., . Then, the manufacturer has more capital to invest in production and greening technology.
Since
, inventory risk is entirely borne by the manufacturer when actual demand is less than the option order quantity of the retailer (scenario 1 in
Table 5). Therefore, from the perspective of avoiding inventory risk, the option contract benefits the retailer. However, due to the limitation of the manufacturer’s production, the retailer cannot replenish any more when actual demand is higher than the production volume of the manufacturer (scenario 2 in
Table 5). Compared with the option contract
, the option contract
reduces the manufacturer’s inventory risk when actual demand is greater than the option order quantity of the retailer and less than the production volume of the manufacturer (scenario 2 in
Table 4).
Proposition 9. , if and only if.
From Proposition 9, we know that when , both the option order quantity of the retailer and the production volume of the manufacturer are equal to the optimal production volume under the centralized supply chain. Meanwhile, the manufacturer’s optimal carbon emission reduction rate with the contract equals that under the centralized system. That means supply chain coordination can be attained with the option contract . There is a negative correlation between option price and strike price under coordination, i.e., , since option price brings a guaranteed revenue for the manufacturer while strike price leads to uncertainty of future earnings (the retailer may not exercise options, or a second order may not happen). If the option price increases, the manufacturer obtains more payments for an option fee. However, the decline in strike price will cause a decline in the manufacturer’s potential future earnings. Therefore, the manufacturer could achieve a delicate balance with an appropriate option order quantity (i.e., ).
Proposition 10. The optimal carbon emission reduction rate of the manufacturer is increasing in carbon emission tax, i.e., . The total profit of the manufacturer is convex in the carbon emission tax .
Substituting
into Equation (7), we have
. When
, the profit of the manufacturer is a decreasing function. When
, the profit of the manufacturer is an increasing function. Therefore, when carbon emission tax increases, total profit of the manufacturer first decreases and then increases as shown in
Figure 4 and
Figure 5.
5.3. The Prepayment-Based Option (PBO) Contract
Under the PBO contract, firstly, the retailer optimizes option order quantity
and preorder quantity
. Then, the manufacturer optimizes production volume
and carbon emission reduction rate
. The cash flow of supply chain with the PBO contract is as follows (see
Figure 8):
: Retailer pays in advance for the preorder quantity and the option order quantity.
: represents the retailer’s expected sales and his total sales revenue would be . The number of options to be exercised is . Meanwhile, the retailer’s transfer payment for exercising options is . is expected sales at wholesale price of the manufacturer. Meanwhile, the transfer payment for the retailer’s second order is .
: The retailer’s expected inventory is while the manufacturer’s expected inventory is . The per-unit salvage value is . Then, the manufacturer pays carbon emission tax .
Then, the retailer’s expected profit will be
In Equation (9), the first term is total sales revenue, the second term is prepayment for preorder quantity, the third term is cost of buying options, the fourth term denotes cost of exercising options, the fifth term is cost of a second order and the last term is total salvage value.
The manufacturer’s expected profit will be
In Equation (10), the first, second, third and fourth terms are revenue from preselling, options selling and exercising and second-selling, respectively. The fifth term is total salvage value. The sixth term is total cost of production and carbon emission tax. The last term is cost of greening technology investment. The manufacturer has to ensure the preorder and the option order quantity. Therefore, her production volume satisfies the constraint .
Two cases are considered: (i) the total order quantity of the retailer is lower than the production volume of the manufacturer, (ii) vice versa. Using the Lagrangian relaxation approach, the manufacturer’s optimal production volume is and her optimal carbon emission reduction rate is when . However, the manufacturer’s optimal production volume is and her optimal carbon emission reduction rate is when . That means the manufacturer’s optimal production volume with the PBO contract is .
Case 1. The total order quantity of the retailer is assumed to be lower than the production volume of the manufacturer. Substituting into Equation (9) when , we will obtain the retailer’s optimal preorder quantity and his optimal option order quantity .
Proposition 11. There exists a prepayment-based option contract . The retailer’s optimal preorder quantity is and his optimal option order quantity is . The manufacturer’s optimal production volume is and her optimal carbon emission reduction rate is . Furthermore, the option price satisfies the following conditions: .
Corollary 7. The manufacturer’s optimal production volume and carbon emission reduction rate with the PBO contract are equal to those under the decentralized system, i.e., , .
Similar to Corollary 4, if the option price can rise without a cap, the PBO contract
cannot change the manufacturer’s production volume and investment of greening technology, i.e.,
, . Whether actual demand is lower or higher than the total order quantity of the retailer (scenario 1, scenario 2 and scenario 3 in
Table 6), the manufacturer always needs to bear inventory risk. The retailer just bears inventory risk when actual demand is less than his preorder quantity (scenario 1 in
Table 6). However, when actual demand is higher than the production volume of the manufacturer, the retailer will not be able to meet demand (scenario 4 in
Table 6).
Case 2. The total order quantity of the retailer is assumed to be greater than the production volume of the manufacturer. Substitute into Equation (9) when . Then, we will obtain the retailer’s optimal preorder quantity and his option order quantity .
Proposition 12. There exists a prepayment-based option contract . The retailer’s optimal preorder quantity is and his optimal option order quantity is . The manufacturer’s optimal production volume is and her optimal carbon emission reduction rate is . Furthermore, the option price satisfies the following conditions: , .
Corollary 8. The manufacturer’s optimal production volume and optimal carbon emission reduction rate with the PBO contract is greater than that under the decentralized system, i.e., , .
Similar to the option contract , reflects the manufacturer’s strategy of the PBO contract. The manufacturer’s option price is decreasing in . When the option price is lower than a certain threshold , the optimal production volume and the optimal carbon emission reduction rate are higher than without a contract, i.e., , . That means the manufacturer’s cash flow is improved by the option fee. The lower option price attracts the retailer to order more options in advance, i.e., and . Then, the manufacturer has more capital to invest in production and greening technology. However, the discount factor does not affect the manufacturer’s strategy of the PBO contract. Moreover, different from the APD contract, the preorder quantity under the PBO contract does not affect the manufacturer’s production volume and investment of greening technology, i.e., .
If actual demand is lower than the preorder quantity of the retailer (scenario 1 in
Table 7), the inventory risk is shared by manufacturer and retailer. The salvage value of the retailer is
while the salvage value of the manufacturer is
. However, inventory risk is entirely borne by the manufacturer, under the single option contract. The retailer entirely bears inventory risk under the APD contract. Therefore, compared with the APD contract and the option contract, the PBO contract
generates intermediate allocations of inventory risk between manufacturer and retailer.
Compared with the decentralized system, inventory risk is fully borne by the manufacturer when actual demand is higher than the retailer’s preorder quantity and lower than the manufacturer’s production volume (scenario 2 in
Table 7). However, the manufacturer also has the salvage value and option fee
.
Proposition 13. , if and only if.
From Proposition 13, we know that when , both the total order quantity of the retailer and the production volume of the manufacturer equal the optimal production volume under the centralized system. Meanwhile, the manufacturer’s optimal carbon emission reduction rate with a contract is equal to that under the centralized system. That means supply chain coordination could be reached with the PBO contract . Similar to Proposition 9, there is a negative relationship between the option price and the strike price under coordination, i.e., .
Proposition 14. The optimal carbon emission reduction rate of the manufacturer is increasing in carbon emission tax, i.e., . The total profit of the manufacturer is convex in the carbon emission tax .
Substituting
into Equation (10), we have
. When
, the profit of the manufacturer is a decreasing function. When
, the profit of the manufacturer is an increasing function. Therefore, when carbon emission tax increases, the total profit of the manufacturer first decreases and then increases, as shown in
Figure 4 and
Figure 5. Compared with the APD contract
, the manufacturer’s total profit with the PBO contract
is first lower as
and then higher as
.